Global well-posedness to the incompressible Navier–Stokes equations with damping

Global well-posedness to the incompressible Navier–Stokes equations with damping

Xin Zhong

School of Mathematics and Statistics, Southwest University Chongqing 400715, People’s Republic of China Received 9 April 2017, appeared 4 September 2017

Communicated by Maria Alessandra Ragusa

Abstract. We study the Cauchy problem of the 3D incompressible Navier–Stokes equa- tions with nonlinear damping termα|u|^β−1u (α > 0 andβ ≥ 1). It is shown that the strong solution exists globally for anyβ≥1.

Keywords: Navier–Stokes equations, global well-posedness, damping.

2010 Mathematics Subject Classification: 35Q35, 35B65, 76N10.

1 Introduction

We are concerned with the following incompressible Navier–Stokes equations with damping inR³:















∂_tu−_µ∆u+u· ∇u+α|u|^β−1u+∇P=0, divu=0,

u(_0,x) =_u0(x)_,

|xlim|→_∞|u(t,x)|=0,

(1.1)

where u= (u¹(t,x),u²(t,x),u³(t,x))is the velocity field, P(t,x)is a scalar pressure. t ≥0 is the time, x ∈ _R³ is the spatial coordinate. In the damping term, α > 0 and β ≥ 1 are two constants. The prescribed functionu₀(x)is the initial velocity field with divu₀= 0, while the constantµ>0 represents the viscosity coefficient of the flow.

This model comes from porous media flow, friction effects, or some dissipative mecha- nisms, mainly as a limiting system from compressible flows (see [1] for the physical back- ground). The problem (1.1) was studied firstly by Cai and Jiu [1], they showed the existence of a global weak solution for any β ≥ 1 and global strong solutions for β ≥ ⁷₂. Moreover, the uniqueness is shown for any ⁷₂ ≤ β ≤ 5. In [8], Zhang et al. proved for β > 3 and u0 ∈ H¹∩L^β+1 that the system (1.1) has a global strong solution and the strong solution is unique when 3 < β ≤ 5. Later, Zhou [9], Gala and Ragusa [3,4], improved the results in

BEmail: xzhong1014@amss.ac.cn

[1,8]. He obtained that the strong solution exists globally for β ≥3 and u₀ ∈ H¹. Moreover, regularity criteria for (1.1) is also established for 1≤ β<3 as follows: ifu(t,x)satisfies

u∈ L^s(0,T;L^γ) with 2 s + ³

γ ≤1, 3<γ<_∞, (1.2) or

∇_u∈ L^˜s(0,T;L^γ˜) with 2 s˜ + ³

γ˜

≤1, 3<γ˜ <_∞, (1.3) then the solution remains smooth on[0,T]. However, the global existence of strong solution to the problem (1.1) for 1≤ β<3 is still unknown. In fact, this is the main motivation of this paper.

Before stating our main result, we first explain the notations and conventions used throughout this paper. We denote by

Z

·dx =

Z

R³ ·dx.

For 1≤ p≤_∞and integerk ≥0, the standard Sobolev spaces are denoted by:

L^p = L^p(_R³), H^k = H^k,2(_R³).

Now we define precisely what we mean by strong solutions to the problem (1.1).

Definition 1.1(Strong solutions). A pair(_u,P)is called a strong solution to (1.1) inR³×(_0,T) if (1.1) holds almost everywhere inR³×(0,T)and

u∈ L^∞(0,T;H¹(_R³))∩L²(0,T;H²(_R³)). Our main result reads as follows.

Theorem 1.2. Suppose that1 ≤ β < ₃ andu0 ∈ H¹(_R³) withdivu0 = 0. Then there exists an absolute constantε₀independent ofu₀,µ,α, andβ, such that if

µ⁻⁴ku0k²_L2

k∇u0k²_L2 + ^α

µ(β+1)ku0k^β+1

L^β+1

≤ ε0, (1.4)

the problem(1.1)has a unique global strong solution.

Remark 1.3. Due to 1 ≤β<3, we get from Hölder’s and Sobolev’s inequalities that ku0k^β+1

L^β+1 ≤ ku0k

5−β 2

L² ku0k

3β−3 2

L⁶ ≤Cku0k

5−β 2

L² k∇u0k

3β−3 2

L² .

Consequently, for the given initial velocityu0 ∈ H¹(_R³)with divu0 =0, it follows from (1.4) that the problem (1.1) has a unique global strong solution when the viscosity constant µ is sufficiently large ork_u0k_L2k∇_u0k_L2 is small enough.

The rest of this paper is organized as follows. In Section 2, we collect some elementary facts and inequalities that will be used later. In Section 3, we show the local existence and uniqueness of solutions of the Cauchy problems (1.1). Finally, we give the proof of Theo- rem1.2in Section4.

2 Preliminaries

In this section, we will recall some known facts and elementary inequalities that will be used frequently later.

We begin with the following Gronwall’s inequality, which plays a central role in proving a priori estimates on strong solutions (u,P).

Lemma 2.1. Suppose that h and r are integrable on (a,b) and nonnegative a.e. in (a,b). Further assume that y∈ C[a,b],y⁰ ∈ L¹(a,b), and

y⁰(t)≤ h(t) +r(t)y(t) for a.e.t∈ (a,b). Then

y(t)≤

y(a) +

Z _t

a h(s)exp

−

Z _s

a r(τ)dτ

ds

exp _{Z t}

a r(s)ds

, t∈[a,b]. Proof. See [7, pp. 12–13].

Next, the following Gagliardo–Nirenberg inequality will be used later.

Lemma 2.2. Let1≤ p,q,r ≤ _{∞, and j,}m are arbitrary integers satisfying0≤ j< m. Assume that v∈C^∞_c (_Rⁿ). Then

kD^jvk_Lp ≤Ckvk¹_L⁻q^akD^mvk^a_Lr, where

−j+ ⁿ

p = (1−a)ⁿ q +a

−m+ⁿ r

, and

a ∈





 h_j

m, 1

, if m−j− ⁿ_r is an nonnegative integer, hj

m, 1i

, otherwise.

The constant C depends only on n,m,j,q,r,a.

Proof. See [5, Theorem].

Finally, we need the following lemma to obtain the uniform bounds in the next section.

Lemma 2.3. Let g∈W^1,1(0,T)and k∈ L¹(0,T)satisfy dg

dt ≤ F(g) +k in[0,T], g(0)≤ g₀,

where F is bounded on bounded sets fromRintoR. Then for everyε >0, there exists T_ε independent of g such that

g(t)≤g₀+ε, ∀t ≤T_ε. Proof. See [6, Lemma 6].

3 Local existence and uniqueness of solutions

In this section, we shall prove the following local existence and uniqueness of strong solutions to the Cauchy problem (1.1).

Theorem 3.1. Suppose that 1 ≤ β < 3, u₀ ∈ H¹(_R³) withdivu₀ = 0. Then there exist a small positive time T₀>₀and a unique strong solution(_u,P)to the Cauchy problem(1.1)inR³×(_0,T₀].

3.1 A priori estimates

The main goal of this subsection is to derive the following key a priori estimates on Φ(t) defined by

Φ(t),ku(t)k²_H1 +1, which are needed for the proof of Theorem3.1.

Proposition 3.2. Assume thatu0 ∈ H¹(_R³)withdivu0 =0. Let(u,P)be a solution to the problem (1.1)on R³×(0,T]. Then there exist a small time T0 ∈ (0,T]and a positive constant C depending only onµ,α,β, and E₀ ,ku₀k_H1+1such that

sup

0<t≤T0

Φ(t)≤C. (3.1)

Proof. Multiplying (1.1)₁ byu and integrating (by parts) the resulting equation over R³, we obtain that

1 2

d dt

Z

|u|²dx+α Z

|u|^β+1dx+µ Z

|∇u|²dx=−

Z

(u· ∇)u·udx. (3.2) By the divergence theorem and (1.1)2, we have

−

Z

(_u· ∇)_u·udx =−

Z

uⁱ∂_iu^ju^jdx=

Z

uⁱ∂_iu^ju^jdx=

Z

(_u· ∇)_u·udx.

Thus

−

Z

(_u· ∇)_u·udx=_{0. (3.3)}

Inserting (3.3) into (3.2) and integrating with respect tot, we get ku(t)k²_L2+α

Z _t

0

kuk^β+1

L^β+1ds+µ Z _t

0

k∇uk²_L2ds≤ ku₀k²_L2. (3.4) Multiplying (1.1)₁ by u_t and integrating (by parts) the resulting equation over R³, we obtain from Cauchy–Schwartz inequality that

µ 2

d dt

Z

|∇_u|²dx+ ^α β+1

d dt

Z

|_u|^β+1dx+

Z

|_ut|²dx=−

Z

(_u· ∇)_u·_utdx

≤ ¹ 2

Z

|ut|²dx+ ¹ 2

Z

|u· ∇u|²dx.

Thus

µd dt

Z

|∇_u|²dx+ ^2α β+1

d dt

Z

|_u|^β+1dx+

Z

|_ut|²dx≤

Z

|_u· ∇_u|²dx. (3.5) Similarly, multiplying (1.1)₁by−_∆uand integrating the resulting equations overR³, we have

1 2

d dt

Z

|∇u|²dx+αβ Z

|u|^β−1|∇u|²dx+µ Z

|_∆u|²dx=

Z

(u· ∇)u·_∆udx

≤ ^µ 2 Z

|_∆u|²dx+ ¹ 2µ

Z

|u· ∇u|²dx.

Hence, we get µd

dt Z

|∇u|²dx+2µαβ Z

|u|^β−1|∇u|²dx+µ² Z

|_∆u|²dx≤

Z

|u· ∇u|²dx, (3.6) which combined with (3.5) yields

2µd dt

Z

|∇u|²dx+ ^2α β+1

d dt

Z

|u|^β+1dx+

Z

|u_t|²dx+2µαβ Z

|u|^β−1|∇u|²dx+µ² Z

|_∆u|²dx

≤2 Z

|u· ∇u|²dx. (3.7)

Applying the Gagliardo–Nirenberg inequality and Sobolev’s inequality, the right hand side of (3.7) can be bounded as

J ,²

Z

|u· ∇u|²dx

≤Ck_uk_L6k_∆uk_L2k∇_uk²_L2

≤Ck_∆uk_L2k∇uk³_L2

≤ ^µ

2

2 k_∆uk²_L2+ ^C

µ²k∇uk⁶_L2. (3.8)

Substituting (3.8) into (3.7), we deduce that d

dtk∇uk²_L2 + ^α µ(β+1)

d dt

Z

|u|^β+1dx+ ¹

2µkutk²_L2 +^µ

4k_∆uk²_L2 ≤ ^C

µ³k∇uk⁴_L2k∇uk²_L2. (3.9) Then integrating (3.9) with respect tot, we have

k∇u(t)k²_L2 ≤C+Cexp

C Z _t

0 Φ(s)²ds

. (3.10)

Combining (3.4) and (3.10), we deduce

Φ(t)≤C+Cexp

C Z _t

0 Φ(s)²ds

. (3.11)

Let us defineΨ(t),Rt

0Φ(s)²ds, then we infer from (3.11) that d

dtΨ(t)≤[C+Cexp(CΨ(t))]².

Hence, the desired (3.1) follows from this inequality and Lemma2.3. This completes the proof of Proposition3.2.

3.2 Proof of Theorem3.1

Since a priori estimates in higher norms have been derived, the local existence of strong solutions can be established by a standard Galerkin method (see for example [2]), and we omit the details. Thus we complete the proof of the existence part of Theorem3.1.

The uniqueness part of Theorem 3.1 is an immediate consequence of the weak-strong unique result in [9, Theorem 3.1]. This finishes the proof of Theorem3.1.

4 Proof of Theorem 1.2

Throughout this section, we denote

C0,ku0k²_L2.

Sometimes we useC(f)to emphasize the dependence on f. Let (u,P)be the strong solution to the problem (1.1) onR³×(0,T), then one has the following results.

Lemma 4.1. For any t∈(0,T), there holds ku(t)k²_L2+α

Z _t

0

kuk^β+1

L^β+1ds+µ Z _t

0

k∇uk²_L2ds≤C₀. (4.1) Proof. This follows from (3.4).

Lemma 4.2. There exists an absolute constant C independent of T,u0,µ,α, and β, such that for any t∈ (0,T), there holds

sup

0≤s≤t

k∇_uk²_L2 ≤ k∇_u0k²_L2+ ^α

µ(β+1)k_u0k^β+1

L^β+1+ ^CC0 µ⁴ sup

0≤s≤t

k∇_uk⁴_{L2. (4.2)} Proof. We infer from (3.7) that

2µd dt

Z

|∇u|²dx+ ^2α β+1

d dt

Z

|u|^β+1dx+

Z

|u_t|²dx+2µαβ Z

|u|^β−1|∇u|²dx+µ² Z

|_∆u|²dx

≤2 Z

|_u· ∇_u|²dx. (4.3)

Then we obtain after integrating (4.3) with respect totthat 2µ sup

0≤s≤t

k∇uk²_L2+ ^2α

β+₁₀^sup_≤s≤tkuk^β+1

L^β+1+µ² Z _t

0

k_∆uk²_L2ds

≤2µk∇u0k²_L2 + ^2α

β+1ku0k^β+1

L^β+1+2 Z _t

0

ku· ∇uk²_L2ds. (4.4) By virtue of the Gagliardo–Nirenberg and Sobolev’s inequalities, one finds that

2ku· ∇uk²_L2 ≤ Ckuk²_L∞k∇uk²_L2

≤ Ck_uk_L6k_∆uk_L2k∇_uk²_L2

≤ Ck_∆uk_L2k∇uk³_L2

≤ ^µ

2

2 k_∆uk²_L2 +Cµ⁻²k∇uk⁶_L2. (4.5) Substituting (4.5) into (4.4) and employing (4.1), we derive that

2µ sup

0≤s≤t

k∇_uk²_L2+ ^2α β+1 sup

0≤s≤t

k_uk^β+1

L^β+1+ ^µ

2

2 Z _t

0

k_∆uk²_L2ds

≤2µk∇u₀k²_L2 + ^2α

β+1ku₀k^β+1

L^β+1 +Cµ⁻² Z _t

0

k∇uk⁶_L2ds

≤2µk∇u₀k²_L2 + ^2α

β+1ku₀k^β+1

L^β+1 +Cµ⁻³ sup

0≤s≤t

k∇uk⁴_L2

Z _t

0 µk∇uk²_L2ds

≤2µk∇u₀k²_L2 + ^2α

β+1ku₀k^β+1

L^β+1 +CC₀µ⁻³ sup

0≤s≤t

k∇uk⁴_L2. This implies the desired (4.2) and finishes the proof of Lemma4.2.

Lemma 4.3. There exists a positive constantε₀independent of T,u₀,µ,α, andβ, such that sup

0≤t≤T

k∇uk²_L2 ≤2k∇u0k²_L2+ ^2α

µ(β+1)ku0k^β+1

L^β+1, (4.6)

provided that

µ⁻⁴C₀

k∇u0k²_L2+ ^α

µ(β+1)ku0k^β+1

L^β+1

≤ε₀. (4.7)

Proof. Define functionE(t)as follows

E(t), ^sup

0≤s≤t

k∇uk²_L2.

In view of the regularity ofu, one can easily check that E(t)is a continuous function on[0,T]. By (4.2), there is an absolute constantM such that

E(t)≤ k∇u₀k²_L2 + ^α

µ(β+1)ku₀k^β+1

L^β+1 +Mµ⁻⁴C₀E²(t). (4.8) Now suppose that

Mµ⁻⁴C₀

k∇u₀k²_L2 + ^α

µ(β+1)ku₀k^β+1

L^β+1

≤ ¹

8, (4.9)

and set

T∗,^max

t∈ [0,T]:E(s)≤4k∇u0k²_L2 + ^4α

µ(β+1)ku0k^β+1

L^β+1, ∀s∈ [0,t]

. (4.10)

We claim that

T∗ =T.

Otherwise, we have T∗ ∈(0,T). By the continuity ofE(t), it follows from (4.8)–(4.10) that E(T∗)≤ k∇u0k²_L2+ ^α

µ(β+1)ku0k^β+1

L^β+1+Mµ⁻⁴C0E²(T∗)

≤ k∇u₀k²_L2+ ^α

µ(β+1)ku₀k^β+1

L^β+1+Mµ⁻⁴C₀E(T∗)

4k∇u₀k²_L2 + ^4α

µ(β+1)ku₀k^β+1

L^β+1

= k∇_u0k²_L2+ ^α

µ(β+1)k_u0k^β+1

L^β+1+4Mµ⁻⁴C₀

k∇_u0k²_L2+ ^α

µ(β+1)k_u0k^β+1

L^β+1

E(T∗)

≤ k∇u0k²_L2+ ^α

µ(β+1)ku0k^β+1

L^β+1+ ¹ 2E(T∗), and thus

E(T∗)≤2k∇u₀k²_L2+ ^2α

µ(β+1)ku₀k^β+1

L^β+1. This contradicts with (4.10).

Choosingε₀ = _8M¹ , by virtue of the claim we showed in the above, we derive that E(t)≤2k∇u₀k²_L2+ ^2α

µ(β+1)ku₀k^β+1

L^β+1, 0<t <T,

provided that (4.7) holds true. This gives the desired (4.6) and consequently completes the proof of Lemma4.3.

Now, we can give the proof of Theorem1.2.

Proof of Theorem1.2. Let ε₀ be the constant stated in Lemma 4.3 and suppose that the initial velocityu0∈ H¹(_R³)with divu0=0, and

µ⁻⁴C0

k∇u0k²_L2+ ^α

µ(β+1)ku0k^β+1

L^β+1

≤ ε₀.

According to Theorem3.1, there is a unique local strong solution (u,P) to the system (1.1).

LetT^∗be the maximal existence time to the solution. We will show that T^∗ = _∞. Suppose, by contradiction, that T^∗ < ∞, then by (1.2), we deduce that for any(s,γ)with ²_s+ _γ³ ≤ 1, 3<

γ<_∞,

Z _T^∗

0

kuk^s_Lγdt= _∞,

which combined with the Sobolev inequalitykuk_L6 ≤ Ck∇uk_L2 leads to Z _T^∗

0

k∇uk⁴_L2dt=_∞. (4.11)

By Lemma4.3, for any 0<T< T^∗, there holds sup

0≤t≤T

k∇_uk²_L2 ≤₂k∇_u0k²_L2+ ^2α

µ(β+1)k_u0k^β+1

L^β+1, which together with Hölder’s and Sobolev’s inequalities implies that

Z _T^∗

0

k∇uk⁴_L2dt≤4

k∇u₀k²_L2+ ^α

µ(β+1)ku₀k^β+1

L^β+1

T^∗

≤4

k∇u0k²_L2+ ^α

µ(β+1)ku0k

5−β 2

L² ku0k

3β−3 2

L⁶

T^∗

≤C(µ,α,β)

k∇u₀k²_L2+ku₀k

5−β 2

L² k∇u₀k

3β−3 2

L²

T^∗

<+_∞,

contradicting to (4.11). This contradiction provides us that T^∗ = _∞, and thus we obtain the global strong solution. This finishes the proof of Theorem1.2.

Acknowledgements

The author would like to express his gratitude to the reviewers for careful reading and helpful suggestions which led to an improvement of the original manuscript. X. Zhong is supported by Fundamental Research Funds for the Central Universities (No. XDJK2017C050), China Postdoctoral Science Foundation (No. 2017M610579), and the Doctoral Fund of Southwest University (No. SWU116033).

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